KABUL—Taliban attacks killed at least 12 people across Afghanistan on Wednesday, underscoring the country’s deteriorating security situation as the U.S.-led combat mission winds down this month.

On Wednesday morning, a suicide bomber on a motorcycle blew himself up at the gate of a bank in Lashkar Gah, the capital of the volatile province of Helmand, local officials said. The explosion breached the entrance to a local branch of New Kabul Bank, allowing three other attackers to enter the building.